Hello, everybody. I'd like to talk about a little problem that has intrigued people ever since we discovered mirrors. Let's have a look at what I see when I look into the bathroom mirror in the morning. Scary. When I point up, Mirror Mathologer points up; when I point down, Mirror Mathologer points down. When I raise my left hand, Mirror Mathologer raises his right hand. When I raise my right hand, Mirror Mathologer raises his left hand. Strange. Also, with writing: The tops of these characters are on top,
both here and for the Mirror Mathologer. And, the bottoms are on the bottom
here and also for Mirror Mathologer. But, this one here runs from left to right,
and the other one runs from right to left. What's going on here? It appears that mirrors reflect left and right
but, not up and down. And in the setup everything's sort of equal. So, if something gets flipped one way, it should also be flipped the other way, right? Now, just to scramble your brain and a little bit further - have a look at this photo here
of a mountain reflected in a mirror lake. Here, up and down is being flipped,
but not left and right. Now if you find all this intriguing,
you are in very good company. I've linked in a video here of one my favorite scientists, Richard Feynman,
talking about him being intrigued by this. Richard Feynman:
Genius.
Nobel laureate.
Really, really famous theoretical physicist. This is a very slippery eel to pin down because... when it comes to words like:
- Flipping
- Mirror Reflection
- Mirror Images ...ordinary language is very "fuzzy". For example, it doesn't distinguish between different phenomena... when it comes to reflection, or mirror images, of 2D objects and 3D objects. And, that's also part of the reason why most of the explanations of this puzzle
(even the really, really good ones)... kind of go off course a little bit,
and maybe in the end confuse more than they explain. So, what I thought I'd do today is try to provide a
(there are many)
really clean explanation. So there's one very important observation which lets us sidestep this whole confusion (2D/3D). and that is: Whatever the explanation, if it works, it has to work for 2D things, like writing here on my t-shirt. And, if it works for 2D things, it automatically works for 3D things. Why? Because I can replace any 3D things
(like myself)
with a photo - which is a flat thing,
and show it to the mirror. Ok, so... A clean explanation of the puzzle should just talk about 2D things. I've prepared a 2D thing. "CAT" written on a piece of paper. What exactly is a mirror image of a 2D thing like this? Well, let's have a look. Look at this thing in the mirror.
There's the mirror image. Now, to really pin this thing down,
let's press the piece of paper against the mirror. Now, the original and the mirror image
are facing the same spots. So, corresponding points are touching each other. So, if I were to make the piece of
paper transparent (except for the writing)... the back of this would actually be a mirror image, right? And, I have actually arranged this
so that it actually works. Now things are
transparent, and you can see... This one object (this one object - this picture frame)
has the original and the mirror image on both sides. So it's really the same thing. And this provides us with a short answer to our puzzle, which is this: When we're saying the mirror exchanges left and right, but not up and down What we're really doing is we're comparing the original to the mirror image. So, we first look at the original. And then, to be able to look at a mirror image, we make a flip... to either show thing to the mirror, or here the physical object So the flip is done by us in some way so that there's there's a flip in there. It's not the mirror; the mirror reflects. Whatever is here on the right comes back straight here. Whatever is on the top comes back straight here. Whatever is here comes back straight here. The mirror doesn't exchange anything. Ok.
That's a short answer, but there's a bit more to it. So, for example:
What about this this mountain reflected in the lake? So what I want to do now is to compare the original
to the mirror image in all possible ways
and see what I can say in general. OK? Now I've prepared a very special object, a 2D object
(it's just a blob)
To take out any sort of bias we might have. I'm not using a cat, or something else recognizable. OK? Now, let's have a look at the mirror image.
There it is. But even this representation has a bit of a bias built in. The way we kind of put these things next to each other
will become very important in a little while. But for the moment let's put this thing down
in an arbitrary way, OK? And now we are comparing original to mirror image. And if you just kind of looked at this for the first time
you may not even see that
there is any similarity between these two things. There is definitely no flipping happening. But when we move things one on top of the other
we get one of those strange Rorschach pictures ...with a nice symmetry axis here. Pull them back apart again and you'll see that
the mirror image differs from the original
by a flip across this distinguished axis. There you go. And that will always be the case. No matter how we juxtapose original and mirror image,
there's always going to be this distinguished axis. So whenever we're comparing original and mirror image, there is a distinguished axis across which we flip to get from one to the other. It's always going to be there. And that, of course, explains most of the puzzle. But we still have to somehow explain:
Why do we get a vertical axis, and not some other axis? But actually... Usually when you look at something via mirror,
you see the thing and a mirror image at the same time. And whenever you do this,
this axis is going to be apparent. So:
The bathroom mirror surface is a bit different from that,
because there you only see the mirror image So let's have a look at the usual set up. You look at something by a mirror and
you see the mirror image and the original. Where the axis is then depends on
the relative position of the mirror and original. So in this case you basically got the mountain and Mirror Lake setup. The mirror axis is going to be down here. And we actually can do the flipping here physically, like so. When I change the position of the mirror,
the mirror axis will change. [It] will always be the bottom part of this mirror frame here. And, we can do the flipping. The flipping happening. And we move things a little bit further. There is a flip like that happening. Now, let's have a look at a special case:
Bathroom Mirror. [In a] Bathroom mirror, we only see the mirror image. Actually, I have to fess up here. I've got a special guest again. The blob was really not a blob.
It was Professor Farnsworth,
one of my favorite characters from Futurama. I'm a huge Futurama fan. So this is what Professor Farnsworth
sees in the mirror in the morning. Ok? Now that's the only thing he sees.
Now, he makes a statement about
"left and right but not up and down" How does the original come into the picture? Well, he (or we) supply it.
We somehow put it in,
virtually, in our minds. How do we put it in? Well, if I give you your original on a piece of paper, like so, would you put it next to that one like that? Probably not.
The way you put it next to the mirror image
is going to be like this. There's a certain bias in which we put the mirror image and original next to each other. It's our bias. It's what we do.
We provide the second image in a certain way. And it's kind of natural. This guy here is basically a person standing up in front of us with a natural up going up. So when we put the original next to that?
Well, we will also put the "up" up.
Because that's how people stand in space. So the explanation for all of this? Whenever you compare original to mirror image,
there's always going to be a flip axis. And for the special case of the bathroom mirror: We arrange things in a certain way. So we arrange for the missing picture to be compared to the original in a certain way,
and that then ensures that we get a vertical axis. Now, often you find explanations of this puzzle culminating in the statements: Mirrors don't exchange left and right;
they don't exchange up and down.
What they do do, is to exchange front and back. And that's a true statement,
but it's actually confusing in this context. As I said before,
you only need really to talk about 2D images. That's essential bits.
As soon as you kind of drag in the 3D bits,
things get confusing. So explanation for the puzzle
is finished now. What comes now is a footnote. Left and right doesn't get flipped?
Well that's reflected by these two arrows here
going in the same direction. Up and down dont get flipped?
That's corresponds by these two red arrows going in the same direction. But what happens to the green arrow - the green arrow that's kind of pointing towards the mirror? The reflection is actually pointing in the opposite direction. Ok, so, in terms of 3D objects (like
myself)? The 3D mirror image is actually only inverted in the direction of the of the green arrow. That's how I turn myself into my mirror image. Also, us being roughly self-similar across this mirror plane here probably contributes to the overall puzzlement that we experience at the very beginning. Now I'm really crazy about mirrors and there would be a lot more things that I could say. What I want to do is... I want to just leave you with a couple of fun things. And for those I've asked my friend, Professor Farnsworth, to model. And here he is standing in front of the straight mirror. Now what I want you to do is... I want you to get a spoon like this and then you look at it in various ways. So for example,
if you look into the spoon something strange happens All of the sudden you see yourself standing on your head. If you look at the spoon like this you'll actually just see a normal mirror image (a little bit distorted). Now for something really spooky, just take the spoon and bring it really close to one of your eyes. Now, you can try and explain those things in the comments. I'm look forward to your answers. Also, if you want something else crazy to experiment with: You need a mirror like this.
I cut this out of a garbage bin. And what a mirror like this does is it actually shows you like other people see you. So, not flipped. So that this thing doesn't show your normal mirror image. It shows you the way other people see you. A little bit distorted in this one.
But basically, that's what it is. So, all very nice effects. Hopefully you've enjoyed this, and hopefully you'll be able to provide some explanations here. That's it for today.
They don’t flip left-right -- if you hold a penny out to your right, then the reflection's penny is also to your right.
But if someone facing you (eg your reflection) is holding a penny in their hand that is to your right, then we’d normally call that “their left” hand. But it’s still to your right.
If someone were facing away from you, ie in the same direction as you, ie not your reflection, and they were holding a penny in their hand that is to your right, we'd call that “their right” hand. Their right and definitely still your right.
I guess you could say that the mirror makes your reflection face in the opposite way that you are facing (flipping away-towards), and that is causing the “left” / “right” hand discrepancy. But in all of these scenarios, the penny is still on your right, so there’s no left-right flip and certainly no up-down flip.
Edits: clarity (hopefully) and typographical
Edit2: P.S. when you point towards your reflection, the reflection points away from your reflection -- flipping away-towards.
Edit3: Toward is a really weird looking word and I have mixed feelings about the fact that it can be spelled either way.
A couple things... left and right depend on orientation. For example, a mirror does not flip east and west if you label left and right that way.
The other thing is Feynman on mirrors.
As we try to place ourselves in the mirror's shoes, we perform a rotation and a flip. Because the z-axis is our preferred axis of rotation, the flip is left-right... we tend not to rotate this way
Why is this better than Feynman's answer?
What a load of obfuscatory crap.
The mirror line/plane is always in the same exact place: it is the surface of the mirror. The direction in which things flip just depends on how you orient the mirror.
The easiest way prove to yourself how any mirror works is just "ray tracing": draw the straights lines from your eye reflecting off various points on its surface and see where they land.
Also all mirroring is equivalent: even a ten dimensional objects has only two reflections (aka chiralities); everything else can be reached by rotation. That is why left-right flipping and front-back flipping can be confused in the first place.
And in fact: for any object and its mirroring in N dimensional space, they are just a rotation of each other in N+1 dimensional space. For example with CAT in 2D you need a flip to get TAƆ but in 3D you can just rotate.
The way I've thought of it is that mirrors reverse chirality - If I take a set of three perpendicular axes like in the video, and then produce a mirror image, I can choose any two axes and make them line up, and the third axis will have been reversed.
"Mirrors exchange front/back but not left/right, up/down" is exactly the right answer modulo a rigid motion that puts the mirror in the XY plane.
Because we're left-right symmetric, we perform a rigid motion in our minds that aligns our right limbs with the mirror image's left limbs and vice versa. But this motion is (obviously) not the same transformation that the mirror does. We could just as well mentally perform a motion that aligns our feet with the mirror image's head but keeps left left and right right.
Sorry if this was already stated in the video, couldn't be bothered to watch.
You and your perfect clone are standing in front of a large pane of glass. Imagine your clone going to the other side and facing you so as to make a mirror image.
.
.
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Why is your clone standing upright? I meant for your clone to go to the other side and stand on his/her head.
"You asshole! Then my clone's feet would align with my head! The mirror image would be wrong!"
But right now your clone's left hand is aligned with your right. That is wrong as well. But don't worry too much about it. Your brain does agree with you.
But it is the best answer: it is just not always the most elucidating.